Discussion:
The identified correlations between the location of the topological defects and the plastic events indicate that these quantities are intimately related to each other. Zones of the sample which have a high density of defects with topological charge \(+1\)  are stable towards shearing since in their vicinity the amplitude of the vibrational excitations are small or vary smoothly in space, i.e., the application of a local strain will not affect significantly these vibrational modes and thus will not give rise to a plastic event. This is also the case for the boundary between two neighboring regions of +1 TD with opposite chirality, since also then the flow field is a smooth function in space, see Fig.  REF(e). For zones with a high density of TD with topological charge \(-1\) , the vector field of the eigen-modes will instead be strongly disturbed by the presence of a local strain, thus making it is highly probable for a plastic event to occur. Note that this connection between TD and PE can be expected to hold only if the local shear is indeed a smooth function in space, i.e., if the sample can locally be described as an elastic medium. This is only the case if the considered length scales are sufficiently large, which in turn rationalizes our finding that the correlation between PE and TD is pronounced if the distance between the TDs is larger than the lengths scale on which the system can support acoustic phonons, while for smaller scales the correlation is lost. Our results also shed light on the previous studies in which machine learning approaches have been used to predict the location of PE [1]}, [2]}, [3]}, [4]}, [5]}, [6]}. Despite the success of these approaches, the understanding on the physical origin of these soft spots has so far not really been elucidated and our findings do now allow us to interpret these zones in terms of a simple physical quantity. Note that in contrast to previous studies, Refs. [7]}, [8]}, [9]}, [10]}, we do not have to determine the normal modes as a function of the strain in order to predict the location of the PE. Instead this location is obtained directly from the normal modes at strain zero, i.e., it is a purely structural quantity in the unperturbed system. This feature should thus permit us to make headway with analytical calculations that aim to describe the yielding of amorphous systems and hence to obtain a better understanding of this important problem. Finally we mention that the approached presented here can be generalized in a straightforward manner to three dimensional systems and it will thus be highly interesting to determine for these cases the correlation between PE and TD.
